If the truth value of (negation p ➡️ q) ➡️ ( p v negation r) is false, then what is the truth value negation p ↔️ r ?

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Here's the Solution to this Question

Here is the condition:

Given

(!p -> q) -> (p V !r) = False

What is the value of

p <-> r

Solution:

p <-> r
(p -> r) ^ (r -> p)
(!p V r) ^ (!r V p)

2.Let's find out truth values of p,q and r (table below). We see that

(!p -> q) -> (p V !r) = False

only when p = False, q = True and r = True.

| p | q | r | !p -> q | p V !r | (!p -> q) -> (p V !r) |

| F | F | F | F | T | T |

| F | F | T | F | F | T |

| F | T | F | T | T | T |

| F | T | T | T | F | F |

| T | F | F | T | T | T |

| T | F | T | T | T | T |

| T | T | F | T | T | T |

| T | T | T | T | T | T |

(!p V r) ^ (!r V p)

which is simplified version of

p <-> r

(!p V r) ^ (!r V p)
(T V T) ^ (F V F)
T ^ F
False

Thus, truth value of

p <-> r

is False.